Efficient joint detection

ABSTRACT

K data signals, or bursts, are transmitted over a shared spectrum in a code division multiple access communication format. A combined signal is received and sampled over the shared spectrum, as a plurality of received vector versions. The combined signal includes the K transmitted data signals. A plurality of system matrices and an associated covariance matrix using codes and estimated impulse responses of the K data signals is produced. Each system matrix corresponds to a received vector version. The system and covariance matrices are extended and approximated as block circulant matrices. A diagonal matrix of each of the; extended and approximated system and covariance matrices are determined by prime factor algorithm—fast Fourier transform (PFA-FFT) without division of the matrix. The received vector versions are extended. A product of the diagonal matrices and the extended received vector versions is taken. An inverse block discrete Fourier transform is performed by a PFA-FFT on a result of the product to produce the estimated data of the K data signals.

CROSS REFERENCE TO THE RELATED APPLICATION(S)

This application is a continuation of U.S. patent application Ser. No.10/644,361 filed Aug. 20, 2003 which claims the benefit of U.S.Provisional Application No. 60/404,561, filed Aug. 20, 2002.

BACKGROUND

FIG. 1 is an illustration of a wireless communication system 10. Thecommunication system 10 has base stations 12 ₁ to 12 ₅ (12) whichcommunicate with user equipments (UEs) 14 ₁ to 14 ₃ (14). Each basestation 12 has an associated operational area, where it communicateswith UEs 14 in its operational area.

In some communication systems, such as frequency division duplex usingcode division multiple access (FDD/CDMA) and time division duplex usingcode division multiple access (TDD/CDMA), multiple communications aresent over the same frequency spectrum. These communications aredifferentiated by their channelization codes. To more efficiently usethe frequency spectrum, TDD/CDMA communication systems use repeatingframes divided into timeslots for communication. A communication sent insuch a system will have one or multiple associated codes and timeslotsassigned to it.

Since multiple communications may be sent in the same frequency spectrumand at the same time, a receiver in such a system must distinguishbetween the multiple communications. One approach to detecting suchsignals is multiuser detection (MUD). In MUD, signals associated withall the UEs 14, are detected simultaneously. For TDD/CDMA systems, oneof the popular MUD techniques is a joint detection technique using blocklinear equalizer (BLE-JD). Techniques for implementing BLE-JD includeusing a Cholesky or an approximate Cholesky decomposition. Theseapproaches have high complexity. The high complexity leads to increasedpower consumption, which at the UE 14 results in reduced battery life.

Accordingly, it is desirable to have computationally efficientapproaches to detecting received data.

SUMMARY

K data signals, or bursts, are transmitted over a shared spectrum in acode division multiple access communication format. A combined signal isreceived and sampled over the shared spectrum, as a plurality ofreceived vector versions. The combined signal includes the K transmitteddata signals. A plurality of system matrices and an associatedcovariance matrix using codes and estimated impulse responses of the Kdata signals is produced. Each system matrix corresponds to a receivedvector version. The system and covariance matrices are extended andapproximated as block circulant matrices. A diagonal matrix of each ofthe extended and approximated system and covariance matrices aredetermined by prime factor algorithm— fast Fourier transform (PFA-FFT)without division of the matrices. The received vector versions areextended. A product of the diagonal matrices and the extended receivedvector versions is taken. An inverse block discrete Fourier transform isperformed by a PFA-FFT on a result of the product to produce theestimated data of the K data signals.

BRIEF DESCRIPTION OF THE DRAWING(S)

FIG. 1 is a wireless communication system.

FIG. 2 is a simplified transmitter and an efficient joint detectionreceiver.

FIG. 3 is an illustration of a communication burst.

FIGS. 4 a and 4 b are a flow chart of a preferred embodiment forefficient joint detection.

FIG. 5 is an illustration of a data burst indicating extended processingareas.

FIG. 6 is a block diagram of a preferred implementation of efficientjoint detection.

FIG. 7 is a simplified receiver having multiple antennas.

FIG. 8 is a simplified receiver sampling the received signal usingfractional sampling.

FIG. 9 is a simplified receiver having multiple antennas and usingfractional sampling.

FIG. 10 is a block diagram of a preferred implementation of efficientjoint detection for fractional sampling or receive diversity.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT(S)

FIG. 2 illustrates a simplified transmitter 26 and receiver 28 usingefficient joint detection in a TDD/CDMA communication system, althoughefficient joint detection is applicable to other systems, such asFDD/CDMA. In a typical system, a transmitter 26 is in each UE 14 andmultiple transmitting circuits 26 sending multiple communications are ineach base station 12. The joint detection receiver 28 may be at a basestation 12, UEs 14 or both.

The transmitter 26 sends data over a wireless radio channel 30. A datagenerator 32 in the transmitter 26 generates data to be communicated tothe receiver 28. A modulation/spreading/training sequence insertiondevice 34 spreads the data with the appropriate code(s) and makes thespread reference data time-multiplexed with a midamble training sequencein the appropriate assigned time slot, producing a communication burstor bursts.

A typical communication burst 16 has a midamble 20, a guard period 18and two data fields 22, 24, as shown in FIG. 3. The midamble 20separates the two data fields 22, 24 and the guard period 18 separatesthe communication bursts to allow for the difference in arrival times ofbursts transmitted from different transmitters 26. The two data fields22, 24 contain the communication burst's data.

The communication burst(s) are modulated by a modulator 36 to radiofrequency (RF). An antenna 38 radiates the RF signal through thewireless radio channel 30 to an antenna 40 of the receiver 28. The typeof modulation used for the transmitted communication can be any of thoseknown to those skilled in the art, such as quadrature phase shift keying(QPSK) or M-ary quadrature amplitude modulation (QAM).

The antenna 40 of the receiver 28 receives various radio frequencysignals. The received signals are demodulated by a demodulator 42 toproduce a baseband signal. The baseband signal is sampled by a samplingdevice 43, such as one or multiple analog to digital converters, at thechip rate of the transmitted bursts. The samples are processed, such asby a channel estimation device 44 and an efficient joint detectiondevice 46, in the time slot and with the appropriate codes assigned tothe received bursts. The channel estimation device 44 uses the midambletraining sequence component in the baseband samples to provide channelinformation, such as channel impulse responses. The channel informationis used by the efficient joint detection device 46 to estimate thetransmitted data of the received communication bursts as soft symbols.

The efficient joint detection device 46 uses the channel informationprovided by the channel estimation device 44 and the known spreadingcodes used by the transmitter 26 to estimate the data of the desiredreceived communication burst(s).

Although efficient joint detection is explained using the thirdgeneration partnership project (3GPP) universal terrestrial radio access(UTRA) TDD system as the underlying communication system, it isapplicable to other systems. That system is a direct sequence widebandCDMA (W-CDMA) system, where the uplink and downlink transmissions areconfined to mutually exclusive timeslots.

The receiver 28 receives a total of K bursts that arrive simultaneously,within one observation interval. For the 3GPP UTRA TDD system, each datafield of a time slot corresponds to one observation interval. For afrequency division duplex (FDD) CDMA system, the received signals arecontinuous, i.e., not in bursts. To handle the continuous signals, FDDsystems divide the received signals into time segments prior to applyingefficient joint detection.

A code used for a k^(th) burst is represented as c^((k)). The K burstsmay originate from K different transmitters or for multi-codetransmissions, less than K different transmitters.

Each data field of a communication burst has a predetermined number oftransmitted symbols, Ns. Each symbol is transmitted using apredetermined number of chips, which is the spreading factor, Q.Accordingly, each data field has Ns×Q chips. After passing through thewireless radio channel, which can introduce a delay spread of up to W−1chips, the observation interval at the receiver is of length Q×Ns+W−1chips.

The symbol response vector b^((k)) as the convolution of the channelresponse vector h^((k)) with the corresponding spreading code c^((k)) isper Equation 1.b^((k))=h^((k))∘c^((k))  Equation 1∘ denotes the convolutional operator. The length of b^((k)) is SF+W−1.

Using the symbol response vectors, the system matrix A is defined as perEquation 2.

The size of the matrix is (N_(s)·SF+W−1)×N_(s)·K. A is a block Toeplitzmatrix.

Block B is defined as per Equation 3.B=[b⁽¹⁾ b⁽²⁾ . . . b^((K))]  Equation 3The received vector sampled at the chip rate can be represented byEquation 4.r=Ad+n  Equation 4

The size of vector r is (N_(s)·SF+W−1) by 1. This size corresponds tothe observation interval.

Data vector d of size N_(s)·K by 1 has the form of Equation 5.d=[d₁ ^(T) d₂ ^(T) . . . d_(N) _(s) ^(T)]^(T)  Equation 5

The sub-vector d_(n) of size K by 1 is composed of the n^(th) symbol ofeach user and is defined as Equation 6.d_(n)=[d_(n) ⁽¹⁾ d_(n) ⁽²⁾ . . . d_(n) ^((K))]^(T), n=1, . . . ,N_(s)  Equation 6

The vector n of size (N_(s)·SF+W−1) by 1 is the background noise vectorand is assumed to be white.

Determining d using an MMSE solution is per Equation 7.d=R⁻¹(A^(H)r)  Equation 7(·)^(H) represents the Hermetian function (complex conjugate transpose).The covariance matrix of the system matrix R for a preferred MMSEsolution is per Equation 8.R=A ^(H) A+σ ² I  Equation 8σ² is the noise variance, typically obtained from the channel estimationdevice 44, and I is the identity matrix.

Using block circulant approximation and block DFT using PFA-FFTs, d inEquation 7 can be determined per Equation 9.

$\begin{matrix}\begin{matrix}{\underset{\_}{d} = {{F\left( R^{- 1} \right)}{F\left( {A^{H}\underset{\_}{r}} \right)}}} \\{= {F^{- 1}\left( {\Lambda^{- 1}\Lambda_{A}{F\left( {\underset{\_}{r}}_{c} \right)}} \right)}}\end{matrix} & {{Equation}\mspace{14mu} 9}\end{matrix}$F(·) and F⁻¹(·) indicate the block-DFT function and the inverseblock-DFT, respectively. The derivation of the block diagonal matrices Λand Λ_(A) is described subsequently. Instead of directly solvingEquation 9, Equation 9 can be solved using the LU decomposition and theforward and backward substitution of the main diagonal block of Λ.Alternately, Equation 9 can be solved using Cholesky decomposition.

FIG. 4 is a flowchart for a preferred method of determining the datavector d using fast joint detection. The system matrix A is determinedusing the estimated channel response vector h^((k)) and the spreadingcode c^((k)) for each burst, 48. R, the covariance matrix of the systemmatrix, is formed, 49. The system matrix A and its covariance matrix Rare extended block square matrices. The extended A is of size D·Q by D·Kand the extended R is of size D·K by D·K, respectively. D is chosen asper Equation 10.

$\begin{matrix}{D \geq \left\lceil {N_{s} + \frac{W - 1}{Q}} \right\rceil} & {{Equation}\mspace{14mu} 10}\end{matrix}$Both extended matrices are approximated to block circulant matrices,A_(c) and R_(c), 50. Because of the extension of A and R, the receivedvector, r, is extended to the vector r_(c) of size D·SF×1 by insertingzeros, 51. The block-diagonal matrix, Λ, is determined by taking a blockDFT using PFA-FFT of the first block column of R_(c), 52.

The block DFT of matched filtering F(A^(H)r) is approximated by F(A_(c)^(H)r_(c)). It is calculated by taking a block DFT using PFA-FFT ofA_(c) and r_(c), 53. Due to the block-diagonal structure of Λ and Λ_(A),the blocks F(d)^((i)), i=1, . . . , D, are of size K by 1 in F(d). Theyare determined by performing on the main diagonal blocks, Λ^((i)) of Λ,LU decomposition, Λ^((i))=L^((i))U^((i)), forward substitution,L^((i))y^((i))=Λ_(A) ^((i)) ^(H) F(r_(c))^((i)), 55, and backwardsubstitution, U^((i))[F(d)]^((i))=y^((i)), 56. L^((i)) is a lowertriangular matrix. U^((i)) is an upper triangular matrix. Λ_((A)) ^((i))is the i^(th) main diagonal block of size SF by K in Λ_(A) andF(r_(c))^((i)) is the i^(th) block of size Q×1 in F(r_(c)). Λ_(A) is theblock DFT using PFA-FFT of the first column of A_(c) and F(r_(c)) is theblock DFT using PFA-FFT of the vector, r_(c). The estimated data vector,d, is determined by an inverse block-DFT of F(d), 57.

Although Equation 9 is a MMSE based solution, fast joint detection canbe applied to other approaches, such as a zero forcing approach as perEquation 11.Rd=(A ^(H) A)d=A ^(H) r  Equation 11As shown in Equation 11, in the zero forcing solution, the σ²I term isdeleted from Equation 8. The following is a derivation for the MMSEsolution, although an analogous derivation can be used for a zeroforcing solution.

To reduce the complexity in determining F(A^(H)r), a block DFT usingPFA-FFT approach taking advantage of the block-Toeplitz structure of Amay be used as shown in Equation 2. First, by repeating B, we extend Ato a block-square matrix of size D·Q×D·K, to use all of the chip symbolsin the observation interval. The extended A is composed of D² blocks ofsize Q×K. The extended A is approximated to the block-circulant matrixA_(c).

A_(c) can be decomposed into three matrices per Equation 12.A_(c)=F_((Q)) ^(H)Λ_(A)F_((K))  Equation 12F_((n))=F {circle around (×)} I_(n) is a block DFT using PFA-FFT matrixof size D·n by D·n. {circle around (×)} denotes a Kronecker product.I_(n) is an identity matrix of size n×n and F is a DFT matrix of sizeD×D, whose elements f_(il), i and l=1, 2, . . . , D are per Equation 13.

$\begin{matrix}{f_{il} = {\frac{1}{\sqrt{D}}{\exp\left( {{- j}\frac{2\;\pi\;{\mathbb{i}}\; l}{D}} \right)}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$D is the length of the DFT and F^(H) F=I. I is an identity matrix ofsize D×D.

The block diagonal matrix Λ_(A) is of size D·Q×D·K and has the form perEquation 14.

$\begin{matrix}{\Lambda_{A} = \begin{bmatrix}\Lambda_{A}^{(1)} & 0 & \cdots & 0 \\0 & \Lambda_{A}^{(2)} & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & \Lambda_{A}^{(D)}\end{bmatrix}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

Each of its entries Λ_(A) ^((i)), i=1, 2, . . . D, is a Q by K block perEquation 15.

$\begin{matrix}{\Lambda_{A}^{(i)} = \begin{bmatrix}\lambda_{1,1}^{({A,i})} & \cdots & \cdots & \lambda_{1,K}^{({A,i})} \\\vdots & ⋰ & \; & \vdots \\\vdots & \; & ⋰ & \vdots \\\lambda_{Q,1}^{({A,i})} & \cdots & \cdots & \lambda_{Q,K}^{({A,i})}\end{bmatrix}} & {{Equation}\mspace{14mu} 15}\end{matrix}$

Alternatively, the main diagonal blocks, Λ_(A) ^((i)), i=1, 2, . . . D,can be computed by, per Equation 16.[Λ_(A) ⁽¹⁾ ^(T) Λ_(A) ⁽²⁾ ^(T) . . . Λ_(A) ^((D)) ^(T) ]^(T)=(F _((Q)) A_(c)(:,1:K))  Equation 16A_(c)(:,1:K) denotes the first block column of A_(c). Namely, the firstK columns of A_(c). F_((SF)) A_(c)(:,1:K) can be calculated by Q·Kparallel non-block DFTs of length D, using PFA-FFTs.

Due to the extension of A, the received vector r is also extended byinserting zeros, becoming vector r_(c) of size D·Q by 1.

Using the above, F(A^(H)r) is approximated to F(A_(c) ^(H)r_(c)). It canbe written as Equation 17.F(A _(c) ^(H) r _(c))=F _((K)) ^(H)Λ_(A) ^(H) F _((Q)) r _(c)  Equation17

The covariance matrix R of size N_(s)·K×N_(s)·K has the block-squarematrix form shown in Equation 18.

$\begin{matrix}{R = \begin{bmatrix}R_{0} & R_{1}^{H} & \cdots & R_{L}^{H} & \cdots & \cdots & \cdots & \cdots & 0 & \cdots & 0 & 0 \\R_{1} & R_{0} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\\vdots & R_{1} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\R_{L} & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\0 & R_{L} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\\vdots & 0 & ⋰ & R_{L} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & 0 & \vdots \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & R_{L}^{H} & 0 \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & R_{L}^{H} \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\0 & 0 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & R_{L} & \cdots & R_{1} & R_{0}\end{bmatrix}} & {{Equation}\mspace{14mu} 18}\end{matrix}$L is defined per Equation 19.

$\begin{matrix}{L = \left\lceil \frac{Q + W - 1}{Q} \right\rceil} & {{Equation}\mspace{14mu} 19}\end{matrix}$Each entry, R_(i), in the R matrix is a K by K block and 0 is a K by Kzero matrix. Due to the size of the extended A, the matrix R is alsoextended to size D·K by D·K by inserting zeros.

The extended R is approximated to a block-circulant matrix, R_(c), ofsize D·K by D·K per Equation 20.

$\begin{matrix}{R_{c} = \mspace{31mu}\begin{bmatrix}R_{0} & R_{1}^{H} & \cdots & R_{L}^{H} & \cdots & \cdots & \cdots & \cdots & 0 & \cdots & R_{2} & R_{1} \\R_{1} & R_{0} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\\vdots & R_{1} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & R_{L} & \vdots \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & R_{L} \\R_{L} & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\0 & R_{L} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\\vdots & 0 & ⋰ & R_{L} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & 0 & \vdots \\R_{L}^{H} & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & R_{L}^{H} & 0 \\\vdots & R_{L}^{H} & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & R_{L}^{H} \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & ⋰ & \vdots & \vdots \\R_{1}^{H} & R_{2}^{H} & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & R_{L} & \cdots & R_{1} & R_{0}\end{bmatrix}} & {{Equation}\mspace{14mu} 20}\end{matrix}$

The block circulant matrix R_(c), is decomposed into three matrices perEquation 21.R_(c)=F_((K)) ^(H)ΛF_((K))  Equation 21F_((K))=F {circle around (×)} I_(K) is a block DFT using PFA-FFT matrixof size D·K×D·K. {circle around (×)} denotes a Kronecker product. I_(K)is an identity matrix of size K×K and F is a DFT matrix of size D×D asdescribed in Equation 13.

The block diagonal matrix Λ of size D·K by D·K has the form per Equation22.

$\begin{matrix}{\Lambda = \begin{bmatrix}\Lambda^{(1)} & 0 & \cdots & 0 \\0 & \Lambda^{(2)} & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & \Lambda^{(D)}\end{bmatrix}} & {{Equation}\mspace{14mu} 22}\end{matrix}$Each of its entries, Λ^((i)), i=1, 2, . . . , D, is a K by K block, perEquation 23A.

$\begin{matrix}{\Lambda^{(i)} = \begin{bmatrix}\lambda_{1,1}^{(i)} & \cdots & \cdots & \lambda_{1,K}^{(i)} \\\vdots & ⋰ & \; & \vdots \\\vdots & \; & ⋰ & \vdots \\\lambda_{K,1}^{(i)} & \cdots & \cdots & \lambda_{K,K}^{(i)}\end{bmatrix}} & {{Equation}\mspace{14mu} 23A}\end{matrix}$Alternatively, the main diagonal blocks, Λ^((i)), i=1, 2, . . . , D, canbe computed per Equation 23B.[Λ⁽¹⁾ ^(T) Λ⁽²⁾ ^(T) . . . Λ^((D)) ^(T) ]^(T)=(F _((K)) R_(c)(:,1:K)),  Equation 23BR_(c)(:,1:K) denotes the first block column of R_(c). Namely, the firstK columns of R_(c). F_((K)) R_(c)(:,1:K) can be calculated by K²parallel non-block DFTs of length D, using PFA-FFTs.

The estimated data vector, d, in Equation 7 is preferably approximatedper Equation 24A.

$\begin{matrix}\begin{matrix}{\underset{\_}{d} = {R^{- 1}A^{H}\underset{\_}{r}}} \\{\approx {R_{c}^{- 1}A_{c}^{H}{\underset{\_}{r}}_{c}}} \\{= {F_{(K)}^{H}\Lambda^{- 1}\Lambda_{A}^{H}F_{(Q)}{\underset{\_}{r}}_{c}}}\end{matrix} & {{Equation}\mspace{14mu} 24A}\end{matrix}$The block diagonal matrix Λ⁻¹ is per Equation 24B.

$\begin{matrix}{\Lambda^{- 1} = \begin{bmatrix}\Lambda^{{(1)}^{- 1}} & 0 & \cdots & 0 \\0 & \Lambda^{{(2)}^{- 1}} & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & \Lambda^{\;{(D)}^{- 1}}\end{bmatrix}} & {{Equation}\mspace{14mu} 24B}\end{matrix}$The inversion of Λ requires an inversion of K×K matrices, Λ^((i)), i=1,2, . . . , D.

Equation 24A can be rewritten as Equation 25.F(d)=Λ⁻¹Λ_(A) ^(H) F(r _(c))  Equation 25F(r_(c)) is per Equations 26A and 26B.F(r _(c))≡F _((Q)) r _(c)  Equation 26AF(d _(c))≡F _((K)) r _(c)  Equation 26BDue to the block-diagonal structure of Λ⁻¹ and Λ_(A) ^(H), Equation 25can be efficiently calculated as follows. The terms of Equation 25 arepartitioned into D blocks, as per Equation 27.

$\begin{matrix}{\begin{bmatrix}{F\left( \underset{\_}{d} \right)}^{(1)} \\{F\left( \underset{\_}{d} \right)}^{(2)} \\\vdots \\{F\left( \underset{\_}{d} \right)}^{(D)}\end{bmatrix} = {{\begin{bmatrix}\Lambda^{{(1)}^{- 1}} & \mspace{11mu} & \; & \; \\\; & \Lambda^{{(2)}^{- 1}} & \; & \; \\\; & \; & {\; ⋰} & \; \\\; & \; & \; & \Lambda^{{(D)}^{- 1}}\end{bmatrix}\begin{bmatrix}\Lambda_{A}^{(1)} & \mspace{11mu} & \; & \; \\\; & \Lambda_{A}^{(2)} & \; & \; \\\; & \; & {\; ⋰} & \; \\\; & \; & \; & \Lambda_{A}^{(D)}\end{bmatrix}}^{H}{\quad\begin{bmatrix}{F\left( {\underset{\_}{r}}_{c} \right)}^{(1)} \\{F\left( {\underset{\_}{r}}_{c} \right)}^{(2)} \\\vdots \\{F\left( {\underset{\_}{r}}_{c} \right)}^{(D)}\end{bmatrix}}}} & {{Equation}\mspace{14mu} 27}\end{matrix}$

Each block in Equation 27 is solved separately, per Equation 28.F(d)^((i))=Λ^((i)) ⁻¹ Λ_(A) ^(H) F(r _(c))^((i))  Equation 28

F(d)^((i)) is a K by 1 vector. Λ^((i)) is a K by K matrix, per Equation22. Λ_(A) ^((i)) is a Q by K matrix, per Equation 14. F(r_(c))^((i)) isa Q by 1 vector and is composed of elements (1+(i−1)Q) through (i·Q) ofF(r_(c)).

To avoid the direct inversion of Λ^((i)), Equation 28 can be solved byusing LU decomposition and forward and backward substitution. Equation28 is rewritten per Equation 29.Λ^((i)) F(d)^((i))=Λ_(A) ^((i)) ^(H) F(r _(c))^((i))  Equation 29Λ^((i)) is decomposed per Equation 30.Λ^((i))=L^((i))U^((i))  Equation 30L^((i)) is a lower triangular matrix and U^((i)) is an upper triangularmatrix.

Using LU decomposition, Equation 28 is represented as, per Equation 31.L ^((i)) U ^((i)) F(d)^((i))=Λ_(A) ^((i)) ^(H) F(r)^((i))  Equation 31F(d)^((i)) in Equation 31 is solved by forward substitution, perEquation 32, and backward substitution, per Equation 33.Forward Substitution: L ^((i)) y ^((i))=Λ_(A) ^((i)) ^(H)F(r)^((i))  Equation 32Backward Substitution: U ^((i)) [F(d)]^((i)) =y ^((i))  Equation 33

Finally, d is determined for all blocks as per Equation 34.

$\begin{matrix}{\underset{\_}{d} = {{F^{- 1}\left( \underset{\_}{d} \right)} = {F^{- 1}\left( \begin{bmatrix}{F\left( \underset{\_}{d} \right)}^{(1)} \\{F\left( \underset{\_}{d} \right)}^{(2)} \\\vdots \\{F\left( \underset{\_}{d} \right)}^{(D)}\end{bmatrix} \right)}}} & {{Equation}\mspace{14mu} 34}\end{matrix}$

FIG. 6 is a block diagram of a preferred implementation of efficientjoint detection in a TDD/CDMA system. Using the received vector, r,r_(c) is formed by inserting zeros, then a block DFT 100 of r_(c) isperformed per Equation 26, to produce F(r_(c)).

Using the received training sequences, the channel impulse responses foreach transmitted burst, h^((k)), is determined by an estimate channelresponse block 102. Using each channelization code, c^((k)) and thechannel impulse response, h^((k)), the system matrix, A, is determinedby compute block matrix A block 104 per Equation 2.

To determine Λ_(A), the system matrix A is extended by extend block 132,to use all received chips in the observation interval. The first blockcolumn of the block-circulant matrix A_(c) is determined by selectingthe first K columns of the extended A matrix by first block column block114. By taking a block DFT using PFA-FFT 118 using PFA-FFT, Λ_(A) isdetermined.

To determine Λ, R is first determined by compute R block 140. For anMMSE solution, R=A^(H)A+σ²I is used; for a zero forcing solution,R=A^(H)A is used. Due to the size of the extended A, R is also extendedby extend block 134. The first block column of the extended R matrix isdetermined by selecting the first K columns of the extended R matrix byfirst block column block 108. The first block column of the extended Rmatrix is circularized by a circularize block column block 110. Itbecomes the first block column of a block-circulant R_(c). By taking ablock DFT using PFA-FFT by block DFT block 112, A is determined.

To efficiently compute the estimated data vector d, Λ_(A), Λ, andF(r_(c)) are divided into blocks Λ_(A) ^((i)), Λ^((i)), andF(r_(c))^((i)), i=1, 2, . . . , D, respectively, exploiting theblock-diagonal structures of Λ_(A) and Λ, by partition block 136. Thecomplex conjugate transpose of Λ_(A) ^((i)), Λ_((A)) ^((i)) ^(H) , isdetermined by a transpose block 130. A multiplier 128 multiplies Λ_(A)^((i)) ^(H) by F(r_(c))^((i)). Λ^((i)) is decomposed using LUdecomposition by a LU decomposition block 126, per Equation 30. Byperforming forward and backward substitution, per Equations 31-33, usingforward and backward substitution blocks 124 and 122, respectively,F(d)^((i)) is determined. By repeating the LU decomposition and forwardand backward substitution D times, F(d) is found. Taking an inverseblock DFT using PFA-FFT of F(d) by block inverse DFT block 120, d isestimated.

FIGS. 7, 8 and 9 are simplified diagrams of receivers applying efficientjoint detection to multiple reception antennas and/or fractional(multiple chip rate) sampling. A receiver 28 with multiple receptionantennas is shown in FIG. 7. Transmitted bursts are received by eachantenna 40 ₁ to 4 _(m) (40). Each antennas' version of the receivedbursts are reduced to baseband, such as by demodulators 42 ₁ to 42 _(m).The baseband signals for each antenna are sampled by sampling devices 43₁ to 43 _(m) to produce a received vector, r₁ to r_(m), for each antenna40. The samples corresponding to the midamble are processed by a channelestimation device 144 to produce channel response matrices, H₁ to H_(m),for each antenna 40. The received data vector, d, is determined by anefficient joint detection device 142 using the received vectors and thechannel response matrices.

A receiver 28 sampling using fraction sampling is shown in FIG. 8.Transmitted bursts are received by the antenna 40. The received burstsare reduced to baseband, such as by a demodulator 42. The basebandsignal is sampled by a sampling device 43 to produce factional samplesas received vectors, r₁ to r_(m). Each received vector represents chiprate samples sampled at a fraction of a chip offset. To illustrate, fortwice the chip rate sampling, two received vectors r₁ and r₂ areproduced. Each of those vectors has samples spaced by half a chip intime. Samples corresponding to the midamble are processed by a channelestimation device 144 to produce channel response matices, H₁ to H_(m),for each set of fractional samples. The received data vector, d, isdetermined by an efficient joint detection device 142 using the receivedvectors and the channel response matrices.

A receiver 28 with multiple reception antennas and using fractionalsampling is shown in FIG. 9. Transmitted bursts are received by eachantenna 40 ₁ to 40 _(i)(40). Each antennas' version of the receivedbursts are reduced to baseband, such as by demodulators 42 ₁ to 42 _(i).The baseband signals for each antenna are sampled by sampling devices 43₁ to 43 _(j) to produce received vectors, r₁ to r_(m). The receivedvectors for each antenna correspond to each multiple of the chip ratesamples. The samples corresponding to the midamble are processed by achannel estimation device 144 to produce channel response matrices, H₁to H_(m), for each antenna's fractional samples. The received datavector, d, is determined by an efficient joint detection device 142using the received vectors and the channel response matrices.

In applying efficient joint detection to either receive diversity,fractional sampling or both, the received communication bursts areviewed as M virtual chip rate received bursts. To illustrate, for twicethe chip rate sampling and two antenna receive diversity, the receivedbursts are modeled as four (M=4) virtual chip rate received bursts.

Each received burst is a combination of K transmitted bursts. Each ofthe K transmitted bursts has its own code. The channel impulse responsevector of the k^(th) out of K codes and the m^(th) out of the M virtualreceived bursts is h^((k,m)). h^((k,m)) has a length W and is estimatedfrom the midamble samples of the burst of the k^(th) code of the m^(th)virtual received burst.

Each of the N data symbols of the burst of the k^(th) code is perEquation 35.d^((k))=[d₁ ^((k)) d₂ ^((k)) . . . d_(N) ^((k))]^(T), 1≦k≦K  Equation 35

The code of the k^(th) burst is per Equation 36.c^((k))=[c₁ ^((k)) c₂ ^((k)) . . . c_(Q) ^((k))]^(T), 1≦k≦K  Equation 36

The symbol response of the k^(th) code's contribution to the m^(th)virtual burst, b^((k,m)) is per Equation 37.b^((k,m))=h^((k,m)){circle around (×)}c^((k))  Equation 37

The length of the symbol response is Q+W−1. Q is the spreading factor.The system matrix, A^((m)), for each m^(th) received burst is perEquation 38.

Each block B^((m)) is of size (Q+W−1) by K and is per Equation 39.B^((m))=└b^((l,m)) b^((2,m)) . . . b^((k,m))┘  Equation 39

The overall system matrix A is per Equation 40.

$\begin{matrix}{A = \begin{bmatrix}A^{(1)} \\A^{(2)} \\\vdots \\A^{(M)}\end{bmatrix}} & {{Equation}\mspace{14mu} 40}\end{matrix}$

As shown in Equation 38, each sub-system matrix A^((m)) is blockToeplitz. The overall received vector of the M virtual bursts is of sizeM(NQ+W−1) and is per Equation 41.r=[r₁ ^(T) r₂ ^(T) . . . r_(M) ^(T)]^(T)  Equation 41The m^(th) received vector r_(m) is of size NQ+W−1 by 1.

Equation 42 is a model for the overall received vector.r=Ad+n  Equation 42n is the noise variance.

Each m^(th) received virtual burst is per Equation 43.r _(m) =A ^((m)) d+n _(m)  Equation 43n_(m) is the noise variance for the m^(th) received virtual burst.

To solve for the data vector d in Equation 42, a block linear equalizerwith either a zero forcing or minimum mean square error (MMSE) approachmay be used per Equation 44.{circumflex over (d)}=R⁻¹A^(H)r  Equation 44R is the covariance matrix.

For a zero forcing solution, R is per Equation 45.

$\begin{matrix}{R = {{\sum\limits_{m = 1}^{M}{A^{{(m)}^{H}}A^{(m)}}} = {A^{H}A}}} & {{Equation}\mspace{14mu} 45}\end{matrix}$

For a MMSE solution, R is per Equation 46.

$\begin{matrix}{R = {{{\sum\limits_{m = 1}^{M}{A^{{(m)}^{H}}A^{(m)}}} + {\sigma^{2}I}} = {{A^{H}A} + {\sigma^{2}I}}}} & {{Equation}\mspace{14mu} 46}\end{matrix}$

The covariance matrix for either the zero forcing or MMSE solution is ablock Toeplitz. To apply a discrete Fourier transform to theblock-Toeplitz A^((m)) matrix, a block-circulant approximation ofA^((m)), A_(c) ^((m)) is used. To make A^((m)) a block-square matrix,A^((m)) is extended. The extended A^((m)) matrix is then approximated toa block circulant matrix A_(c) ^((m)).

The A_(c) ^((m)) matrix is composed of D by D blocks. Each block is ofsize Q by K. Accordingly, the size of A_(c) ^((m)) becomes DQ by DK. Toinclude all the elements of A^((m)), D is chosen to be an integer largerthan D_(min) as determined per Equation 47.

$\begin{matrix}{D_{\min} = \left\lceil {N + \frac{\left( {W - 1} \right)}{Q}} \right\rceil} & {{Equation}\mspace{14mu} 47}\end{matrix}$┌·┐ represents a round up to an integer function.

The covariance matrix R is a block-square matrix of size NK by NK withblocks of size K by K. For R to be compatible with the extended A_(c)^((m)) matrix, R is extended to the size DK by DK by zero-padding andapproximating the extended R to a block circulant covariance matrixR_(c). For the received vector, r^((m)), to be compatible with A_(c)^((m)) and R_(c), r^((m)) is extended to a DQ by 1 vector, r_(c) ^((m))by zero padding.

After extending the received vectors, r^((m)), the overall receivedvector is per Equation 48.r_(c)=[r_(c) ⁽¹⁾ ^(T) r_(c) ⁽²⁾ ^(T) . . . r_(c) ^((M)) ^(T)]^(T)  Equation 48

Each block-circulant matrix A_(c) ^((m)) is diagonalized to ablock-diagonal matrix by block discrete Fourier transform matrices perEquation 49.A_(c) ^((m))=F_((Q)) ^(H)Λ_(A) ^((m))F_((K))  Equation 49

F_((Q)) is per Equation 50.F_((Q))=F{circle around (×)}I_(Q)  Equation 50

F_((K)) is per Equation 51.F_((K))=F{circle around (×)} I_(K)  Equation 51F is a discrete Fourier transform matrix of size D by D and is an n by nidentity matrix. Λ_(A) ^((m)) is a block diagonal matrix of the form ofEquation 52.

$\begin{matrix}{\Lambda_{A}^{(m)} = \begin{bmatrix}\Lambda_{A}^{({1,m})} & 0 & \cdots & 0 \\0 & \Lambda_{A}^{({2,m})} & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & \Lambda_{A}^{({D,m})}\end{bmatrix}} & {{Equation}\mspace{14mu} 52}\end{matrix}$

Λ_(A) ^((l,m)) for l=1, . . . , D, and m=1, . . . , M is a non-zeroblock of size Q by K. 0 is a zero matrix of size Q by K having all zeroelements.

Λ_(A) ^((l,m)) is alternately computed per Equation 53.Λ_(A) ^((m))=diag^(B)(F _((Q)) A _(c) ^((m))(:,1:K))  Equation 53

A_(c) ^((m))(:,1:K) is the first block column of A_(c) ^((m)). The firstblock column having K columns. To determine Λ_(A) ^((m)), preferablyF_((Q)) A_(c) ^((m))(:,1:K) is determined by QK parallel non-block DFTsof length D, using PFA-FFTs. The block circulant matrix R_(c) is alsopreferably diagonalized to the block diagonal matrix Λ_(R) by a blockDFT matrix F_((K))=F {circle around (×)} I_(K) as per Equation 54.R_(c)=F_((K)) ^(H)Λ_(R)F_((K))  Equation 54

The block diagonal matrix Λ_(R) is composed by blocks Λ_(R) ^((l)), l=1,. . . , D of size K by K in its main diagonal block, per Equation 55.

$\begin{matrix}{\Lambda_{R} = \begin{bmatrix}\Lambda_{R}^{(1)} & 0 & \cdots & 0 \\0 & \Lambda_{R}^{(2)} & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & \Lambda_{R}^{(D)}\end{bmatrix}} & {{Equation}\mspace{14mu} 55}\end{matrix}$

Another approach to determine Λ_(R) is per Equation 56.Λ_(R)=diag^(B)(F _((K)) R _(c)(:,1:K))  Equation 56R_(c)(:,1:K) is the first block column of R_(c). F_((K)) R_(c)(:,1:K) ispreferably determined using K² parallel non-block DFTs of length D. Inone implementation, the K² parallel non-block DFTs are implemented usingK² parallel non-block prime factor algorithm fast Fourier transforms(PFA-FFTs) of length D.

Preferably to perform the block equalization of Equation 44, the matchedfiltering is approximated per Equation 57.

$\begin{matrix}{A^{H}\underset{\_}{r}\begin{matrix}{{\approx {A_{c}^{H}{\underset{\_}{r}}_{c}}} = {\sum\limits_{m = 1}^{M}{A^{{(m)}^{H}}{\underset{\_}{r}}_{c}^{(m)}}}} \\{= {\sum\limits_{m = 1}^{M}{\left( {F_{(Q)}^{H}\Lambda_{A}^{(m)}F_{(K)}} \right)^{H}{\underset{\_}{r}}_{c}^{(m)}}}} \\{= {F_{(K)}^{H}{\sum\limits_{m = 1}^{M}{\Lambda_{A}^{{(m)}^{H}}F_{(Q)}{\underset{\_}{r}}_{c}^{(m)}}}}}\end{matrix}} & {{Equation}\mspace{20mu} 57}\end{matrix}$

The block-diagonalization of A_(c) ^((m)) is per Equation 58.A_(c)=└A_(c) ⁽¹⁾ ^(T) A_(c) ⁽²⁾ ^(T) . . . A_(c) ^((M)) ^(T)┘^(T)  Equation 58

The estimation of the data vector, {circumflex over (d)}, is perEquation 59.

$\begin{matrix}\begin{matrix}{\hat{\underset{\_}{d}} = {{R^{- 1}A^{H}\underset{\_}{r}} \approx {R_{c}^{- 1}A_{c}^{H}{\underset{\_}{r}}_{c}}}} \\{= {\left( {F_{(K)}^{H}\Lambda_{R}F_{(K)}} \right)^{- 1}F_{(K)}^{H}{\sum\limits_{m = 1}^{M}\left( {\Lambda_{A}^{{(m)}^{H}}F_{(Q)}{\underset{\_}{r}}_{c}^{(m)}} \right)}}} \\{= {F_{(K)}^{H}\Lambda_{R}^{- 1}F_{(K)}F_{(K)}^{H}{\sum\limits_{m = 1}^{M}\left( {\Lambda_{A}^{{(m)}^{H}}F_{(Q)}{\underset{\_}{r}}_{c}^{(m)}} \right)}}} \\{= {F_{(K)}^{H}\Lambda_{R}^{- 1}{\sum\limits_{m = 1}^{M}\left( {\Lambda_{A}^{{(m)}^{H}}F_{(Q)}{\underset{\_}{r}}_{c}^{(m)}} \right)}}} \\{= {F_{(K)}^{H}\underset{\_}{y}}}\end{matrix} & {{Equation}\mspace{20mu} 59}\end{matrix}$

The vector y is of size DK by 1 and is per Equation 60.

$\begin{matrix}\begin{matrix}{\underset{\_}{y} = {\Lambda_{R}^{- 1}{\sum\limits_{m = 1}^{M}\left( {\Lambda_{A}^{(m)}F_{(Q)}{\underset{\_}{r}}_{c}^{(m)}} \right)}}} \\{= \left\lbrack {{\underset{\_}{y}}^{{(1)}^{T}}\mspace{14mu}{\underset{\_}{y}}^{{(2)}^{T}}\mspace{14mu}\ldots\mspace{14mu}{\underset{\_}{y}}^{{(D)}^{T}}} \right\rbrack^{T}}\end{matrix} & {{Equation}\mspace{20mu} 60}\end{matrix}$y^((l)), l=1, . . . D is a vector of size K by 1.

Preferably to determine y, F_((Q)) r_(c) ^((m)) is determined using Qparallel non-block DFTs of a length D. In one implementation, the Qparallel non-block DFTs are implemented using Q parallel non-blockPFA-FFTs of length D. Λ_(R) ⁻¹ is a block diagonal matrix having blocksof size K by K in the main diagonal and is per Equation 61.

$\begin{matrix}{\Lambda_{R}^{- 1} = \begin{bmatrix}\Lambda_{R}^{{(1)}^{- 1}} & 0 & \ldots & 0 \\0 & \Lambda_{R}^{{(2)}^{- 1}} & \ldots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \ldots & \Lambda_{R}^{{(D)}^{- 1}}\end{bmatrix}} & {{Equation}\mspace{20mu} 61}\end{matrix}$Each Λ_(R) ^((l)−1), l=1, . . . , D, is a block of size K by K.

Preferably using the block diagonal structure of Λ_(R) ⁻¹, y^((l)) isdetermined by the Cholesky decomposition of Λ_(R) ^((l)) and forward andbackward substitution in parallel. Alternately, Λ_(R) ^((l)) is directlyinverted.

To perform the Cholesky decomposition, a vector

$\sum\limits_{m = 1}^{M}\left( {\Lambda_{A}^{{(m)}^{H}}F_{(Q)}{\underset{\_}{r}}_{c}^{(m)}} \right)$is divided into D blocks of a size K by 1, per Equation 62.

$\begin{matrix}\begin{matrix}{\underset{\_}{x} = {\sum\limits_{m = 1}^{M}\left( {\Lambda_{A}^{{(m)}^{H}}F_{(Q)}{\underset{\_}{r}}_{c}^{(m)}} \right)}} \\{= \left\lbrack {{\underset{\_}{x}}^{{(1)}^{T}}\mspace{14mu}{\underset{\_}{x}}^{{(2)}^{T}}\mspace{14mu}\ldots\mspace{14mu}{\underset{\_}{x}}^{{(D)}^{T}}} \right\rbrack^{T}}\end{matrix} & {{Equation}\mspace{20mu} 62}\end{matrix}$A Cholesky factor G^((l)) of Λ_(R) ^((l)) is determined using afactorization, per Equation 63.Λ_(R) ^((l))=G^((l))G^((l)) ^(H)   Equation 63Using the Cholesky factor G^((l)), each y^((l)) is determined by forwardand backward substitution separately per Equations 64, 65 and 66.Λ_(R) ^((l))y^((l))=G^((l))G^((l)) ^(H) y^((l))=x^((l))  Equation 64Forward Substitution: Find z^((l)) in G^((l))z^((l)=x) ^((l)), wherez^((l))=G^((l)) ^(H) y^((l))  Equation 65Backward Substitution: Find y^((l)) in z^((l))=G^((l)) ^(H)y^((l))  Equation 66

By performing a block inverse DFT of y, the data vector d is estimatedas {circumflex over (d)}. Preferably, the block inverse DFT isimplemented using K parallel non-block inverse PFA-FFTs of a length D.

FIG. 10 is a block diagram of a preferred implementation of efficientjoint detection in a TDD/CDMA system. Although FIG. 10 illustrates usingtwo sets of samples, the figure can be extended to other multiple sets.Using the received vector for each set of chip rate samples, r₁ and r₂,r_(c) ^((l)) and r_(c) ⁽²⁾ is formed by inserting zeros, by Extendblocks 232 ₁ and 232 ₂, respectively. A block DFT 200 ₁ and 200 ₂ usingPFA-FFT of r_(c) ⁽¹⁾ and r_(c) ⁽²⁾ is then performed, to produce F_((Q))r_(c) ⁽¹⁾ and F_((Q)) r_(c) ⁽²⁾.

Using the received training sequences, the channel impulse responses foreach chip rate version of each transmitted burst, h^((k)) ⁽¹⁾ andh^((k)) ⁽²⁾ , is determined by estimate channel response blocks 202 ₁and 202 ₂. Using each channelization code, c^((k)) and the channelimpulse response, h^((k)) ⁽¹⁾ and h^((k)) ⁽²⁾ , each system matrix, A⁽¹⁾and A⁽²⁾, is determined by compute sub-system matrix blocks 204 ₁ and204 ₂ per Equations 37 and 38.

To determine Λ_(A) ⁽¹⁾ and Λ_(A) ⁽²⁾, each system matrix, A⁽¹⁾ and A⁽²⁾,is extended by extend blocks 231 ₁ and 231 ₂. The first block column ofeach block-circulant matrix, A⁽¹⁾ and A⁽²⁾, is determined by selectingthe first K columns of the extended A^((m)) matrix by first block columnblocks 214 ₁ and 214 ₂. By taking block DFTs 218 ₁ and 218 ₂, Λ_(A) ⁽¹⁾and Λ_(A) ⁽²⁾ are determined using a PFA-FFT.

To determine Λ_(R), the first block column of R is determined by computefirst block column R block 240. The first column of R is extended byextend block 234. The first block column of the extended R is determinedby a first block column determining device 208. The first block columnof the extended R matrix is circularized, R_(c), by a circularize blockcolumn block 210. By taking a block DFT by block DFT block 212, Λ_(R) ⁽⁾ is determined using a PFA-FFT.

To efficiently compute the estimated data vector d, each of Λ_(A) ⁽¹⁾,Λ_(A) ⁽²⁾ and F(r_(c) ⁽¹⁾) and F(r_(c) ⁽²⁾) as well as Λ_(R) are used.Each of F(r_(c) ⁽¹⁾), F(r_(c) ⁽²⁾), Λ_(A) ⁽¹⁾, Λ_(A) ⁽²⁾, and Λ_(R) isdivided into D blocks, by partition block 236. The complex conjugatetranspose of each of Λ_(A) ^((1,i)) and Λ_(A) ^((2,i)), Λ_(A) ^((1,i))^(H) and Λ_(A) ^((2,i)) ^(H) , where i is the i^(th) block, isdetermined by transpose blocks 230, 231. A multiplier 228 multipliesΛ_(A) ^((2,i)) ^(H) by F_((Q))(r_(c) ⁽²⁾)^((i)). A multiplier 229multiplies Λ_(A) ^((1,i)) ^(H) by F_((Q))(r_(c) ⁽¹⁾)^((i)). A summer 225sums the multiplied results per Equation 62. Λ_(R) ^((i)) is decomposedusing Cholesky decomposition 226, per Equation 63. By performing forwardand backward substitution, per Equations 65 and 66, using forward andbackward substitution blocks 224 and 222, respectively, F(d) isdetermined. Taking an inverse block DFT using PFA-FFT of F(d) by blockinverse DFT block 220 using a PFA-FFT, {circumflex over (d)} isestimated.

1. An apparatus for solving a linear equation of the form r=A d+n for dusing either a minimum mean square error block linear equalizer(MMSE-BLE) or zero forcing block linear equalizer (ZF-BLE) basedsolution, where A is a block Toeplitz matrix and n represents a noisevector, r and d are vectors, the apparatus comprising: a covariancematrix processing component configured to produce a covariance matrix Rof the form A^(H)A+σ²I for a MMSE-BLE based solution, where A^(H) is aHermetian of A, σ² is a noise variance and I is an identity matrix orA^(H)A for a ZF-BLE based solution; a first matrix extension componentconfigured to extend the A matrix and R matrix; a matrix blockprocessing component configured to approximate the extended A and Rmatrices as block circulant matrices; a diagonal matrix processingcomponent configured to determine a diagonal matrix of each of theextended and approximated A and R matrices, using the block columns ofthe extended and approximated A and R matrices; a second matrixextension component configured to extend r; a block Fourier transformcomponent configured to take a Fourier transform of r; a multipliercomponent configured to take products of the diagonal matrices and theextended r; a summer configured to sum the products; and a block inverseFourier transform component configured to estimate d using an inverseFourier transform and the summed products.
 2. The apparatus of claim 1wherein the first and second matrix extension components are configuredto respectively extend the A and R matrices and r to be compatible witha prime factor algorithm fast Fourier transform.
 3. The apparatus ofclaim 1 further comprising a LU decomposition component configured toperform LU decomposition on the diagonal of the R matrix and; a forwardsubstitution component and a backward substitution component configuredto produce an inverse Fourier transform of d.
 4. The apparatus of claim1 further comprising a Cholesky decomposition component for performingCholesky decomposition on the diagonal of the R matrix and a forwardsubstitution component and a backward substitution component forproducing an inverse Fourier transform of d.
 5. A method for detectingdata from K data signals transmitted over a shared spectrum in a codedivision multiple access format wherein a combined signal over theshared spectrum that includes the K transmitted data signals is receivedand sampled to produce a received vector, a system matrix and anassociated covariance matrix using codes and estimated impulse responsesof the K data signals is produced, the covariance matrix is extended andapproximated as a block circulant matrix, a diagonal matrix of theextended and approximated covariance matrix is determined using a blockcolumn of the extended and approximated covariance matrix, the receivedvector is extended and a Fourier transform thereof is taken, the methodcomprising: conducting sampling to produce a plurality of receivedvector versions; for each vector version, producing a system matrixversion using codes and estimated impulse responses of the K datasignals and determining for each system matrix version a respectivediagonal matrix version based on an extended and approximated blockcirculant version of the respective system matrix version; producing thesystem matrix by combining the system matrix versions produced withrespect to the plurality of received vector versions from which thecovariance matrix is produced; extending each received vector versionand taking a Fourier transform of each; producing a plurality ofproducts such that, for each of the received vector versions, a productis taken of: the Fourier transform of the extended received vectorversion, the diagonal matrix derived from the covariance matrix, and thereceived vector version's respective diagonal matrix version; summingthe plurality of products; and estimating data of the K data signalsusing an inverse Fourier transform of the summed products.
 6. The methodof claim 5 wherein the Fourier transforms are performed using a primefactor algorithm fast Fourier transform.
 7. The method of claim 5wherein each received vector version corresponds to a differentreception antenna.
 8. The method of claim 5 wherein the combined signalis sampled at a multiple of a chip rate of the K data signals and eachreceived vector version corresponds to a different chip rate multiple.9. The method of claim 5 further comprising partitioning the diagonalmatrices into a plurality of partitions.
 10. The method of claim 5wherein the estimating data of the K data signals includes performing LUdecomposition, forward substitution and backward substitution.
 11. Themethod of claim 5 wherein the estimating data of the K data signalsincludes performing Cholesky decomposition, forward substitution andbackward substitution.
 12. A communication station configured to detectdata from K data signals transmitted over a shared spectrum in a codedivision multiple access format having a sampling component configuredto sample a received combined signal over the shared spectrum thatincludes the K transmitted data signals to produce a received vector, acompute sub-system matrix component configured to produce a systemmatrix, a compute covariance matrix component configured to produce anassociated covariance matrix using codes and estimated impulse responsesof the K data signals, an extension component configured to extend thecovariance matrix and a block column component configured toapproximated the extended covariance matrix as a block circulant matrix,a block Fourier transform component configured to determine a diagonalmatrix of the extended and approximated covariance matrix using a blockcolumn of the extended and approximated covariance matrix, the extensioncomponent configured to extend the received vector and the block Fouriertransform component configured to take a Fourier transform thereof, thecommunication station wherein: the sampling component is configured toproduce a plurality of received vector versions; the compute sub-systemmatrix component is configured to produce a system matrix version foreach vector version using codes and estimated impulse responses of the Kdata signals and to combine the system matrix versions; the blockFourier transform component is configured to determine a respectivediagonal matrix version for each vector version based on an extended andapproximated block circulant version of the respective system matrixversion; the extension component is configured to extend each receivedvector version and the block Fourier transform component is configuredto take a Fourier transform of each extended received vector version;and comprising a multiplier component configured to take products suchthat, for each of the received vector versions, a product is taken of:the Fourier transform of the extended received vector version, thediagonal matrix derived from the covariance matrix and the receivedvector version's respective diagonal matrix version; a summer configuredto sum the plurality of products; and a block inverse Fourier transformcomponent configured to estimate data of the K data signals using aninverse Fourier transform of the summed products.
 13. The communicationstation of claim 12 wherein the block Fourier transform component isconfigured to perform Fourier transforms using a prime factor algorithmfast Fourier transform.
 14. The communication station of claim 12further comprising a plurality of antennas associated with the samplingcomponent such that each received vector version corresponds to adifferent reception antenna.
 15. The communication station of claim 12wherein the sampling component is configured such that the combinedsignal is sampled at a multiple of a chip rate of the K data signals andeach received vector version corresponds to a different chip ratemultiple.
 16. The communication station of claim 12 further comprising apartitioning component configured to partition the diagonal matrixversions into a plurality of partitions.
 17. The communication stationof claim 12 wherein the block inverse Fourier transform component isconfigured to estimate data of the K data signals by performing LUdecomposition, forward substitution and backward substitution.
 18. Thecommunication station of claim 12 wherein the block inverse Fouriertransform component is configured to estimate data of the K data signalsby performing Cholesky decomposition, forward substitution and backwardsubstitution.
 19. The communication station of claim 12 configured as abase station.
 20. The communication station of claim 12 configured as auser equipment.